Chemistry Reference
In-Depth Information
aggregate of atomic nuclei and electrons. Its Hamiltonian is invariant under the
following types of transformation:
1. Any permutation of the positions and spins of the electrons.
2. Any rotation of the positions and spins of all particles (electrons and nuclei)
about any axis through the center of mass.
3. Any overall translation in space.
4. The reversal of all particle momenta and spins (time reversal).
5. The simultaneous inversion of the positions of all particles in the center of mass
(parity inversion).
6. Any permutation of the positions and spins of any set of identical nuclei.
The complete (symmetry) group of the Hamiltonian is thus the direct product of
several groups [
246
]. In the context of this Chapter, the operations 5 and 6 are most
important since they correspond to symmetry transformations of the nuclear frame-
work of the molecule, i.e., its geometry and dynamic stereochemistry. The spin of
the particles may be neglected for the present purpose. Furthermore, the very small
(10
20
a.u.) parity-violating weak neutral current interaction lifting the exact
degeneracy of enantiomers of a chiral molecule is neglected [
263
,
264
].
The number of permutations in the complete nuclear permutation-inversion
group is increasing rapidly with the number of identical particles. Fortunately,
most of the permutations leaving the Hamiltonian invariant may be disregarded
from a dynamic stereochemistry point of view because they involve breaking and
forming of bonds. Thus, the corresponding barriers are too high for such processes
to be observed under the experimental conditions in consideration. Only
feasible
permutations and permutation-inversions are included in the molecular symmetry
group [
246
,
247
,
252
].
The concept of feasible permutation-inversions introduces some level of arbi-
trariness and a dependence on the particular experimental conditions or computa-
tional model. On the other hand, the criterion of feasibility is a particularly elegant
and transparent way of introducing the necessary chemical information, i.e., the
constitution (topology), the conformational flexibility and/or chemical reactivity,
into the rigorous mathematical formalism of group theory [
247
]. For example, the
effect of breaking symmetry by substitution may be analyzed in this way. Likewise,
one may add or omit certain operators corresponding to a specific conformational
isomerization process to the feasible permutation-inversion operators of the molec-
ular symmetry group and thus analyze their effect on the dynamic stereochemistry.
3.4 Transition State Symmetry
The transition state is a characteristic point on the pathway between educts and
products in chemical reactions including conformational isomerizations. The transi-
tion state is defined as the lowest possible energy maximum (barrier) on a continuous
pathway from educts to products (assuming a barrier does exist) [
265
-
268
].
